Deterministic Automaton and Logically Definable Sets of Numbers

30 janvier 2017

Arthur Milchior (LACL)

For a fixed base b, any integer can be encoded as a finite word of alphabet of digits. In dimension d>0, a vector of d integers is encoded as a word of alphabet of vector of d digits. A set of vector of integers is thus encoded as a language whose alphabet is the set of vector of digits. Thus, an automaton whose alphabet is the set of vector of digits recognizes a set of integers. Similarly, a Büchi automaton recognizes a set of vector of real.

It is then natural to consider algorithms which decide whether the set of vectors of numbers accepted by a finite automaton admits some properties. For example, Honkala proved in 1986 that it is decidable whether an automaton recognize a FO[N,+,<]-definable set of integers. Muchnik proved a similar result for automata recognizing sets of vectors of reals. A polynomial-time algorithm was then given by Leroux in 2006, and a quasi-linear time algorithm for the case of dimension 1 was given in 2013 by Marsault and Sakarovitch.

We state that it is decidable in linear time:
-whether a set of reals recognized by a given finite minimal weak Büchi automaton is FO[R,+,<]-definable.
-whether a set of vectors recognized by a minimal finite deterministic automaton can be defined in some logics less expressive than FO[N,+], such as FO[N,<,mod].
Furthermore, formulas which defines those sets can be computed in linear time and cubic time respectively.

Furthermore, is shown that it is decidable whether a set of vector of real or of integers accepted by a (weak Büchi) automaton:
-is definable in a logic which admits quantifier-elimination. For example, if the set is definable in FO[R,+,<], FO[R,Z,+,<,mod,floor] where mod is the set of modular predicate, FO[N,<,mod] or FO[<].
-satisfies a first-order formula in some formalism. For example, whether a set is a submonoid/subsemigroup of (R^d,+)

In this talk, I intend to:
-introduce automata recognizing set of vector of numbers,
-characterize the set of numbers which are FO[N,<,mod]- and FO[R,+,<]-definable,
-introduce and generalizes the methods used by Honkala, Muchnik and Marsault-Sakarovitch
-explain how how those methods can be applied to the above-mentioned problems.